Two triangles are congruent if one can be moved (by translation, rotation, and/or reflection) to lie exactly on top of the other. It is a more precise way of saying that two triangles are "the same".
When deciding whether or not two triangles are congruent, we don't need to know six pieces of information (three sides and three angles), we need only three pieces in one of the following combinations.
Side-side-side congruence (SSS)
If two triangles have three equal side lengths, then the triangles must be congruent. Try this yourself with three straight objects - once you put them together, you can rotate, translate, and reflect the triangle to make every other possible combination:
Each triangle is made from the same three sides, so they are all congruent.
This kind of congruence is called side-side-side, or SSS.
Side-angle-side congruence (SAS)
If two triangles have a pair of matching sides and the angles between them are equal, then the triangles must be congruent. Try this yourself with two straight objects - if you hold them together at one end and form an angle, there is only one triangle you can form by joining the ends together:
After fixing the two given sides about the given angle, there is only one possible triangle.
This kind of congruence is called side-angle-side, or SAS. We write this test with the "A" in between the two "S"s, because the angle must be between the matching sides - the long name for this kind of congruence is "two sides and the included angle".
There is no SSA
It is possible for triangles to have two pairs of equal sides and a pair of matching angles, yet not be congruent overall. Here is an example.
This can only happen when the pair of equal angles is not included between the sides.
Try using this applet to find the two different triangles that have two matching angles and a matching non-included angle, just like the picture above:
In order for the two triangles to be congruent, the third corresponding side must be equal. This means that SSS and SAS congruence hold, but not SSA.
Right angle-hypotenuse-side congruence (RHS)
If two right-angled triangles have equal lengths hypotenuses and another pair of equal sides, then the triangles must be congruent:
There is only one possible triangle that lines up the hypotenuse with the given side at a right angle.
Notice that the right angle is not included between the sides - this is the only exception to the general rule, which is why we mention it as a separate case. This congruence test is called right angle-hypotenuse-side, or RHS.
Angle-angle-side congruence (AAS)
What if we are only given one pair of equal sides? In this case we need two pairs of equal angles. Here is the construction if the two angles are made with the given side:
The projected lines meet at exactly one point, so we can only build one triangle from this information.
If one of the given angles is opposite the given side, we can always find the third one by using the angle sum of a triangle:
This kind of congruence is called angle-angle-side congruence, or AAS. To use AAS to show that two triangles are congruent, the matching sides must have the same position relative to the matching angles.
If two triangles don't satisfy these tests, one of two things could be true:
The two triangles are definitely not congruent, or
We don't have enough information to know whether or not they're congruent
Two triangles are definitely not congruent if there is a pair of sides or angles that are in the same relative position but are not equal to each other.
Consider the following:
Which two of the following triangles are congruent?
What congruence test does this pair satisfy?
Consider the following diagram:
Are the triangles \triangle ABD and \triangle CDB definitely congruent?
What congruence test does this pair satisfy?
Select the three statements that, when put together, establish congruence for this test.
Make sure each reason is correct as well.
These are the four congruence tests in proving congruence in triangles:
Side-side-side, or SSS: Three pairs of equal sides.
Side-angle-side, or SAS: Two pairs of equal sides with an equal included angle
Angle-angle-side, or AAS: Two pairs of equal angles and one pair of equal sides
Right angle-hypotenuse-side, or RHS: Both have right angles, equal hypotenuses, and another equal side